This is similar to how a fraction of integers can always be written uniquely in lowest terms by canceling out common factors. However, since F is a unique factorization domain, there is a unique representation for any rational expression P/ Q with P and Q polynomials of lowest degree and Q chosen to be monic. P/ Q is equivalent to R/ S, for polynomials P, Q, R, and S, when PS = QR. Any rational expression can be written as the quotient of two polynomials P/ Q with Q ≠ 0, although this representation isn't unique. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F. In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. This is useful in solving such recurrences, since by using partial fraction decomposition we can write any proper rational function as a sum of factors of the form 1 / ( ax + b) and expand these as geometric series, giving an explicit formula for the Taylor coefficients this is the method of generating functions.Ībstract algebra and geometric notion 4.2 Notion of a rational function on an algebraic varietyĪ function f ( x ) Ĭonversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series.4 Abstract algebra and geometric notion.

Rational expressions